Now we review the literature on similar Hele-Shaw limits. These Hele-Shaw limits are well studied when \(m\)  is a constant and until recently, these limits were only investigated when the system is under congestion with \(m\)  constant or when \(f=\vec{b}=0\)  with \(m(x,t)\)  either being constant or \(1/t\) . The weak limit of \((\ref {PME})\)  when \(m \equiv 1\) , \(f=0\) , and \(\vec{b}=0\)  was studied in [1]} and was again studied in [2]} with more general initial data. This investigation was continued in [3]} and [4]} when the equations \((\ref {PME})\)  were posed on a domain \(\Omega \times (0,\infty )\)  with \(\Omega \)  being a smooth domain with general initial data and pressure boundary data. And in [4]} the authors considered the case of \(m(t)=1/t\)  of \((\ref {Hele-Shaw})\)  in their analysis. Then in [6]}, the author showed uniqueness and existence considered when \(m\)  is constant with \(f\)  and \(\vec{b}=0\)  for viscosity solutions. In addition, in [7]} and [8]} the authors dealt with the Hele-Shaw limits with the condition of \(f>0\) , \(\vec{b}=0\) , and \(m \equiv 1\)  in a weak and viscosity sense respectively. And in [9]} the authors were able to incorporate a non-zero drift when \(\vec{b}(x,t) = \nabla \Phi (x)\)  such that \(\Delta \Phi > 0\)  with \(f=0\)  and \(m \equiv 1\)  into their analysis by using viscosity solutions and optimal transport theory. Then in [10]} the authors were able to deal with the case of \(f + \nabla \cdot \vec{b} > 0\)  by using viscosity solutions.
